3.1717 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3} \]

[Out]

(-2*a^2*(a + b/x)^(7/2))/(7*b^3) + (4*a*(a + b/x)^(9/2))/(9*b^3) - (2*(a + b/x)^
(11/2))/(11*b^3)

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Rubi [A]  time = 0.0796185, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/x^4,x]

[Out]

(-2*a^2*(a + b/x)^(7/2))/(7*b^3) + (4*a*(a + b/x)^(9/2))/(9*b^3) - (2*(a + b/x)^
(11/2))/(11*b^3)

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Rubi in Sympy [A]  time = 10.2944, size = 49, normalized size = 0.83 \[ - \frac{2 a^{2} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{3}} + \frac{4 a \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{3}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{11}{2}}}{11 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(5/2)/x**4,x)

[Out]

-2*a**2*(a + b/x)**(7/2)/(7*b**3) + 4*a*(a + b/x)**(9/2)/(9*b**3) - 2*(a + b/x)*
*(11/2)/(11*b**3)

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Mathematica [A]  time = 0.0429404, size = 47, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (8 a^2 x^2-28 a b x+63 b^2\right )}{693 b^3 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)/x^4,x]

[Out]

(-2*Sqrt[a + b/x]*(b + a*x)^3*(63*b^2 - 28*a*b*x + 8*a^2*x^2))/(693*b^3*x^5)

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Maple [A]  time = 0.007, size = 44, normalized size = 0.8 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-28\,abx+63\,{b}^{2} \right ) }{693\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x^4,x)

[Out]

-2/693*(a*x+b)*(8*a^2*x^2-28*a*b*x+63*b^2)*((a*x+b)/x)^(5/2)/b^3/x^3

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Maxima [A]  time = 1.42421, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}}}{11 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a}{9 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)/x^4,x, algorithm="maxima")

[Out]

-2/11*(a + b/x)^(11/2)/b^3 + 4/9*(a + b/x)^(9/2)*a/b^3 - 2/7*(a + b/x)^(7/2)*a^2
/b^3

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Fricas [A]  time = 0.226791, size = 96, normalized size = 1.63 \[ -\frac{2 \,{\left (8 \, a^{5} x^{5} - 4 \, a^{4} b x^{4} + 3 \, a^{3} b^{2} x^{3} + 113 \, a^{2} b^{3} x^{2} + 161 \, a b^{4} x + 63 \, b^{5}\right )} \sqrt{\frac{a x + b}{x}}}{693 \, b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)/x^4,x, algorithm="fricas")

[Out]

-2/693*(8*a^5*x^5 - 4*a^4*b*x^4 + 3*a^3*b^2*x^3 + 113*a^2*b^3*x^2 + 161*a*b^4*x
+ 63*b^5)*sqrt((a*x + b)/x)/(b^3*x^5)

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Sympy [A]  time = 9.8642, size = 1073, normalized size = 18.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(5/2)/x**4,x)

[Out]

-16*a**(27/2)*b**(9/2)*x**8*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079
*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*
x**(11/2)) - 40*a**(25/2)*b**(11/2)*x**7*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**
(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**
(11/2)*b**10*x**(11/2)) - 30*a**(23/2)*b**(13/2)*x**6*sqrt(a*x/b + 1)/(693*a**(1
7/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13
/2) + 693*a**(11/2)*b**10*x**(11/2)) - 236*a**(21/2)*b**(15/2)*x**5*sqrt(a*x/b +
1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/
2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 1010*a**(19/2)*b**(17/2)*x*
*4*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2)
+ 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 1776*a**(17/
2)*b**(19/2)*x**3*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)
*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2))
- 1570*a**(15/2)*b**(21/2)*x**2*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) +
2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b
**10*x**(11/2)) - 700*a**(13/2)*b**(23/2)*x*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*
x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*
a**(11/2)*b**10*x**(11/2)) - 126*a**(11/2)*b**(25/2)*sqrt(a*x/b + 1)/(693*a**(17
/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/
2) + 693*a**(11/2)*b**10*x**(11/2)) + 16*a**14*b**4*x**(17/2)/(693*a**(17/2)*b**
7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 69
3*a**(11/2)*b**10*x**(11/2)) + 48*a**13*b**5*x**(15/2)/(693*a**(17/2)*b**7*x**(1
7/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(1
1/2)*b**10*x**(11/2)) + 48*a**12*b**6*x**(13/2)/(693*a**(17/2)*b**7*x**(17/2) +
2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b*
*10*x**(11/2)) + 16*a**11*b**7*x**(11/2)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a*
*(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**
(11/2))

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GIAC/XCAS [A]  time = 0.272934, size = 365, normalized size = 6.19 \[ \frac{2 \,{\left (924 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4}{\rm sign}\left (x\right ) + 4851 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b{\rm sign}\left (x\right ) + 11781 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{2}{\rm sign}\left (x\right ) + 16863 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{3}{\rm sign}\left (x\right ) + 15345 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{4}{\rm sign}\left (x\right ) + 9009 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{5}{\rm sign}\left (x\right ) + 3311 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{6}{\rm sign}\left (x\right ) + 693 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{7}{\rm sign}\left (x\right ) + 63 \, b^{8}{\rm sign}\left (x\right )\right )}}{693 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)/x^4,x, algorithm="giac")

[Out]

2/693*(924*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*sign(x) + 4851*(sqrt(a)*x - sqr
t(a*x^2 + b*x))^7*a^(7/2)*b*sign(x) + 11781*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^
3*b^2*sign(x) + 16863*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^3*sign(x) + 15
345*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^4*sign(x) + 9009*(sqrt(a)*x - sqrt(a
*x^2 + b*x))^3*a^(3/2)*b^5*sign(x) + 3311*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^
6*sign(x) + 693*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^7*sign(x) + 63*b^8*sig
n(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^11